# Definition:Dedekind Completion

## Definition

Let $S$ be an ordered set.

A Dedekind completion of $S$ is a Dedekind complete ordered set $\tilde S$ together with an order embedding $\phi: S \to \tilde S$, subject to:

For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \to X$, there exists a unique order embedding $\tilde f: \tilde S \to X$ such that:
$\tilde f \circ \phi = f$

## Also see

This concept is not to be confused with the Dedekind-MacNeille completion.

## Source of Name

This entry was named for Julius Wilhelm Richard Dedekind.